α-Well-Posedness for Quasivariational Inequality Problems

Abstract

We introduce and study the concepts of α-well-posedness and L-α-well-posedness for quasivariational inequality problems having a unique solution and the concepts of α-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense for quasivariational inequality problems having more than one solution. We present some necessary and/or sufficient conditions for the various kinds of well-posedness to occur. Our results generalize and strengthen previously known results for quasivariational inequality problems.

1. Introduction

Let 𝐸 be a reflexive real Banach space and let 𝐾 be a nonempty closed convex subset of 𝐸. Let 𝑆 be a set-valued mapping from 𝐾 to 𝐾 and let 𝐴 be an operator from 𝐸 to the dual space 𝐸^∗. Bensoussan and Lions [1], Baiocchi and Capelo [2], and Mosco [3] considered the following quasivariational inequality (in short, (QVIP)), which is to find a point 𝑢₀∈𝐾 such that𝑢₀∈𝑆𝑢₀,𝐴𝑢₀,𝑢₀−𝑣≤0,∀𝑣∈𝑆𝑢₀. (1.1)

The interest in quasivariational inequality problems lies in the fact that many economic or engineering problems are modeled through them, as explained in [4, 5] where a wide bibliography on variational inequalities, quasivariational inequality problems, and related problems is contained. Moreover, under suitable assumptions, a quasivariational inequality is equivalent to a generalized Nash equilibrium problem [3].

On the other hand, well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution [6]. The study of well-posedness for scalar minimization problems started from Tikhonov [7] and Levitin and Polyak [8]. Since the convergence of numerical methods for quasivariational inequality Problems can be obtained also with the aid of well-posedness theory, Lignola [9] introduced and investigated the concepts of well-posedness and L-well-posedness for quasivariational inequalities having a unique solution and the concepts of well-posedness and L-well-posedness in the generalized sense for quasivariational inequality problems having more than one solution.

In this paper, inspired by the above concepts of well-posedness for (QVIP), we introduce and study the concepts of 𝛼-well-posedness and L-𝛼-well-posedness for quasivariational inequality Problems having a unique solution and the concepts of 𝛼-well-posedness in the generalized sense and L-𝛼-well-posedness in the generalized sense for quasivariational inequality Problems having more than one solution. The results in this paper generalize and improve the results in [9, 10].

2. Preliminaries

Denote by Γ the solution set of (QVIP). Let 𝛼>0. In order to investigate the 𝛼-well-posed for (QVIP), we need the following definitions.

First we recall the notion of Mosco convergence [11]. A sequence (𝐻_𝑛)_𝑛 of subsets of 𝐸 Mosco converges to a set 𝐻 if𝐻=liminf_𝑛𝐻_𝑛=𝑤−limsup_𝑛𝐻_𝑛, (2.1) where lim_𝑛inf𝐻_𝑛 and 𝑤−lim_𝑛sup𝐻_𝑛 are, respectively, the Painlevé-Kuratowski strong limit inferior and weak limit superior of a sequence (𝐻_𝑛)_𝑛, that is,liminf_𝑛𝐻_𝑛=𝑦∈𝐸∶∃𝑦_𝑛∈𝐻_𝑛,𝑛∈𝑁,with𝑦_𝑛⟶𝑦,𝑤−limsup_𝑛𝐻_𝑛=𝑦∈𝐸∶∃𝑛_𝑘↑+∞,𝑛_𝑘∈𝑁,∃𝑦_𝑛𝑘∈𝐻_𝑛𝑘,𝑘∈𝑁,with𝑦_𝑛𝑘⇀𝑦, (2.2) where “⇀” means weak convergence, “→” means strong convergence.

If 𝐻=lim_𝑛inf𝐻_𝑛, we call the sequence (𝐻_𝑛)_𝑛 of subsets of 𝐸 Lower Semi-Mosco which converges to a set 𝐻.

It is easy to see that a sequence (𝐻_𝑛)_𝑛 of subsets of 𝐸 Mosco converges to a set 𝐻 which implies that the sequence (𝐻_𝑛)_𝑛, also Lower Semi-Mosco, converges to the set 𝐻, but the converse is not true in general.

We will use the usual abbreviations usc and lsc for “upper semicontinuous” and “lower semicontinuous,” respectively. Let 𝐸 be a reflexive real Banach space with dual 𝐸^∗. An operator 𝐴∶𝐸→𝐸^∗ will be called hemicontinuous if it is continuous from every segment of 𝐸 to 𝐸^∗ endowed with the weak topology. 𝐴∶𝐸→𝐸^∗ will be called monotone if ⟨𝐴𝑢−𝐴𝑣,𝑢−𝑣⟩≥0 for every 𝑢, and 𝑣∈𝐸. 𝐴∶𝐸→𝐸^∗ will be called pseudomonotone if ⟨𝐴𝑢,𝑢−𝑣⟩≤0⇒⟨𝐴𝑣,𝑢−𝑣⟩≤0 for every 𝑢 and 𝑣∈𝐸.

Definition 2.1. A sequence (𝑢_𝑛)_𝑛 is an 𝛼-approximating sequence for (QVIP) if
(i)(𝑢_𝑛)∈𝐾, for all 𝑛∈𝑁;(ii)there exists a sequence (𝜀_𝑛)_𝑛,𝜀_𝑛>0, decreasing to 0 such that𝑑𝑢_𝑛,𝑆𝑢_𝑛≤𝜀_𝑛,thatis,𝑢_𝑛∈𝐵𝑆𝑢_𝑛,𝜀_𝑛,∀𝑛∈𝑁,𝐴𝑢_𝑛,𝑢_𝑛−𝛼−𝑣2𝑢_𝑛−𝑣²≤𝜀_𝑛,∀𝑣∈𝑆𝑢_𝑛,∀𝑛∈𝑁. (2.3)

Definition 2.2. A quasivariational inequality (QVIP) is said to be 𝛼-well-posed (resp., 𝛼-well-posed in the generalized sense) if it has a unique solution 𝑢₀ and every 𝛼-approximating sequence (𝑢_𝑛)_𝑛 strongly converges to 𝑢₀ (resp., if the solution set Γ of (QVIP) is nonempty and for every 𝛼-approximating sequence (𝑢_𝑛)_𝑛 has a subsequence which strongly converges to a point of Γ).

Definition 2.3. A sequence (𝑢_𝑛)_𝑛 is an L-𝛼-approximating sequence for (QVIP) if:
(i)(𝑢_𝑛)∈𝐾, for all 𝑛∈𝑁;(ii)there exists a sequence (𝜀_𝑛)_𝑛,𝜀_𝑛>0, decreasing to 0 such that 𝑑(𝑢_𝑛,𝑆(𝑢_𝑛))≤𝜀_𝑛, and𝐴𝑣,𝑢_𝑛−𝛼−𝑣2𝑢_𝑛−𝑣²≤𝜀_𝑛,∀𝑣∈𝑆𝑢_𝑛,∀𝑛∈𝑁. (2.4)

Definition 2.4. A quasivariational inequality (QVIP) is said to be L-𝛼-well-posed (resp., L-𝛼-well-posed in the generalized sense) if it has a unique solution 𝑢₀ and every L-𝛼-approximating sequence (𝑢_𝑛)_𝑛 strongly converges to 𝑢₀ (resp., if the solution set Γ of (QVIP) is nonempty and for every L-𝛼-approximating sequence (𝑢_𝑛)_𝑛 has a subsequence which strongly converges to a point of Γ).
It is worth noting that if 𝛼=0, then the definitions of 𝛼-well-posedness, 𝛼-well-posedness in the generalized sense, L-𝛼-well-posedness, and L-𝛼-well-posedness in the generalized sense for (QVIP), respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for (QVIP) in [9]. We also note that Definition 2.2 generalizes and extends 𝛼-well-posedness and 𝛼-well-posedness in the generalized sense of variational inequalities in [10] which are related to the continuously differentiable gap function of variational inequality Problems introduced by Fukushima [12].

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.5 (see [13]). Let (𝐻_𝑛)_𝑛 be a sequence of nonempty subsets of the space 𝐸 such that
(i)𝐻_𝑛 is convex for every 𝑛∈𝑁;(ii)𝐻₀⊆lim_𝑛inf𝐻_𝑛;(iii)there exists 𝑚∈𝑁 such that int⋂_𝑛≥𝑚𝐻_𝑛≠∅.
Then, for every 𝑢₀∈int𝐻₀, there exists a positive real number 𝛿 such that 𝐵(𝑢₀,𝛿)⊆𝐻_𝑛, for all 𝑛≥𝑚.
If 𝐸 is a finite dimensional space, then assumption (iii) can be replaced by
(iii)’int𝐻₀≠∅.

The following Lemmas 2.6 and 2.7 play important roles in this paper. Now we present a Minty type lemma for quasivariational inequalities as follows.

Lemma 2.6. Suppose that set-valued mapping 𝑆 is nonempty convex-valued, the operator 𝐴 is hemicontinuous and monotone, 𝑢₀∈𝑆(𝑢₀). Then the following conditions are equivalent:
(i)⟨𝐴𝑢₀,𝑢₀−𝑣⟩−(𝛼/2)‖𝑢₀−𝑣‖²≤0, for all 𝑣∈𝑆(𝑢₀),(ii)⟨𝐴𝑣,𝑢₀−𝑣⟩−(𝛼/2)‖𝑢₀−𝑣‖²≤0, for all 𝑣∈𝑆(𝑢₀).

Proof. We first prove that (ii) implies (i). Let 𝑣 be a arbitrary point of 𝑆(𝑢₀). For every number 𝑡∈[0,1], since the set-valued mapping 𝑆 is convex-valued and 𝑢₀∈𝑆(𝑢₀), the point 𝑣_𝑡=𝑡𝑣+(1−𝑡)𝑢₀ belongs to 𝑆(𝑢₀). It follows from (ii) that 𝐴𝑣_𝑡,𝑢₀−𝑣_𝑡−𝛼2𝑢₀−𝑣_𝑡²≤0. (2.5) From the definition of 𝑣_𝑡, one has lim_𝑡→0𝐴𝑣_𝑡,𝑢₀−𝛼−𝑣2𝑡𝑢₀−𝑣²≤0, (2.6)and it follows from the hemicontinuity of 𝐴 that 𝐴𝑢₀,𝑢₀−𝑣≤0, (2.7) then 𝐴𝑢₀,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²≤0,∀𝑣∈𝑆𝑢₀. (2.8) The converse is an easy consequence of monotonicity of 𝐴.

Lemma 2.7. Assume that set-valued mapping 𝑆 is nonempty convex-valued, then 𝑢₀∈Γ if and only if the following conditions hold:𝑢₀∈𝑆𝑢₀,𝐴𝑢₀,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²≤0,∀𝑣∈𝑆𝑢₀. (2.9)

Proof. The necessity is clearly held. Now we prove the sufficiency. Let for all 𝑣∈𝑆(𝑢₀), for all 𝑡∈[0,1], 𝑣_𝑡=𝑡𝑣+(1−𝑡)𝑢₀. Since 𝑆 is convex-valued, 𝑣_𝑡∈𝑆(𝑢₀), one has 𝐴𝑢₀,𝑢₀−𝑣_𝑡−𝛼2𝑢₀−𝑣_𝑡²≤0,∀𝑡∈(0,1], (2.10) which implies that 𝐴𝑢₀,𝑢₀𝛼−𝑣−𝑡2𝑢₀−𝑣²≤0,∀𝑡∈(0,1],∀𝑣∈𝑆𝑢₀. (2.11) The above inequality implies, for 𝑡 converging to zero, that 𝑢₀ is a solution of (QVIP). This completes the proof.

3. Case of a Unique Solution

In this section, we investigate some metric characterizations of 𝛼-well-posedness and 𝐿-𝛼-well-posedness for (QVIP).

For any 𝜀>0, we consider the set𝑄_𝜀=(𝑆(𝑢)),⟨⟩−𝛼𝑢∈𝐾∶𝑢∈𝐵,𝜀𝐴𝑢,𝑢−𝑣2‖𝑢−𝑣‖²(𝑢𝐿≤𝜀,∀𝑣∈𝑆)_𝜀=(𝑆(𝑢)),⟨⟩−𝛼𝑢∈𝐾∶𝑢∈𝐵,𝜀𝐴𝑣,𝑢−𝑣2‖𝑢−𝑣‖²≤𝜀,∀𝑣∈𝑆(𝑢). (3.1)

Theorem 3.1. Let the same assumptions be as in Lemma 2.7. Then, one has(a)(QVIP) is 𝛼-well-posed if and only if the solution set Γ of (QVIP) is nonempty and lim_𝜀→0diam𝑄_𝜀=0;(b)moreover, if 𝐴∶𝐸→𝐸^∗ is pseudomonotone, then (QVIP) is 𝐿-𝛼-well-posed if and only if the solution set Γ of (QVIP) is nonempty and lim_𝜀→0diam𝐿_𝜀=0.

Proof. We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (QVIP) is 𝛼-well-posed, then Γ≠∅. It follows from Lemma 2.7 that 𝑄_𝜀≠∅. Suppose by contradiction that there exists a real number 𝛽, such that lim_𝜀→0diam𝑄_𝜀>𝛽>0, then there exists 𝜀_𝑛>0, with 𝜀_𝑛→0, and (𝑤_𝑛)_𝑛,(𝑧_𝑛)_𝑛∈𝑄_𝜀𝑛, such that ‖𝑤_𝑛−𝑧_𝑛‖>𝛽, for all 𝑛∈𝑁. Since the sequences (𝑤_𝑛)_𝑛,(𝑧_𝑛)_𝑛 are both 𝛼-approximating sequences for (QVIP), (𝑤_𝑛)_𝑛 and (𝑧_𝑛)_𝑛 strongly converge to the unique solution 𝑢₀, and this gives a contradiction. Therefore, lim_𝜀→0diam𝑄_𝜀=0.
Conversely, let (𝑢_𝑛)_𝑛,𝑢_𝑛∈𝐾, be an 𝛼-approximating sequence for (QVIP). Then there exists a sequence 𝜀_𝑛>0, with 𝜀_𝑛→0, such that 𝑑𝑢_𝑛,𝑆𝑢_𝑛≤𝜀_𝑛,∀𝑛∈𝑁,𝐴𝑢_𝑛,𝑢_𝑛−𝛼−𝑣2𝑢_𝑛−𝑣²≤𝜀_𝑛,∀𝑣∈𝑆𝑢_𝑛,∀𝑛∈𝑁. (3.2) that is, 𝑢_𝑛⊂𝑄_𝜀𝑛, for all 𝑛∈𝑁. It is easy to see lim_𝜀→0diam𝑄_𝜀=0 and Γ≠∅ implying that Γ is a singleton point set. Indeed, if there exist two different solutions 𝑧₁,𝑧₂, then from Lemma 2.7, we know that 𝑧₁,𝑧₂∈𝑄_𝜀, for all 𝜀>0. Thus, lim_𝜀→0diam𝑄_𝜀≥‖𝑧₁−𝑧₂‖≠0, a contraction. Let 𝑢₀ be the unique solution of (QVIP). It follows from Lemma 2.7 that 𝑢₀∈𝑄_𝜀𝑛. Thus, lim_𝑛→0‖𝑢_𝑛−𝑢₀‖≤lim_𝑛→0diam𝑄_𝜀𝑛=0. So (𝑢_𝑛)_𝑛 strongly converge to 𝑢₀. Therefore, (QVIP) is 𝛼-well-posed.

Theorem 3.2. Let 𝛼>0 and the following assumptions hold:
(i)the set-valued mapping 𝑆 is nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converges to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑢₀);(ii)for every converging sequence (ℎ_𝑛)_𝑛, there exists 𝑚∈𝑁, such thatint_𝑛≥𝑚𝑆ℎ_𝑛≠∅; (3.3)(iii)the operator 𝐴 is hemicontinuous and monotone on 𝐾.Then, (QVIP) is 𝛼-well-posed if and only if 𝑄_𝜀≠∅,∀𝜀>0,lim_𝜀→0diam𝑄_𝜀=0. (3.4)

Proof. The necessity has been proved in Theorem 3.1(a).
Conversely, assume that (3.4) holds. It is easy to see that (3.4) implies that the solution set Γ of (QVIP) is a singleton point set. Let (𝑢_𝑛)_𝑛 be an 𝛼-approximating sequence for (QVIP), that is, there exists a sequence 𝜀_𝑛>0, with 𝜀_𝑛→0, such that 𝑑𝑢_𝑛,𝑆𝑢_𝑛≤𝜀_𝑛,∀𝑛∈𝑁,𝐴𝑢_𝑛,𝑢_𝑛−𝛼−𝑣2𝑢_𝑛−𝑣²≤𝜀_𝑛,∀𝑣∈𝑆𝑢_𝑛,∀𝑛∈𝑁. (3.5) Therefore, 𝑢_𝑛⊂𝑄_𝜀𝑛, for all 𝑛∈𝑁. In light of (3.4), (𝑢_𝑛)_𝑛 is a Cauchy sequence and strongly converges to a point 𝑢₀∈𝐾. In order to obtain that 𝑢₀ solves (QVIP), we start to prove that 𝑢₀∈𝑆(𝑢₀). For each 𝑛∈𝑁, choose 𝑢^′_𝑛∈𝑆(𝑢_𝑛), such that ‖𝑢_𝑛−𝑢^′_𝑛‖<𝑑(𝑢_𝑛,𝑆(𝑢_𝑛))+𝜀_𝑛≤2𝜀_𝑛. It follows from 𝑢_𝑛→𝑢₀ and 𝜀_𝑛→0 that 𝑢^′_𝑛→𝑢₀. It follows from the assumption (i) that lim_𝑛inf𝑆(𝑢_𝑛)=𝑆(𝑢₀). Thus, 𝑢₀∈𝑆(𝑢₀).
To complete the proof, consider an arbitrary point 𝑣∈𝑆(𝑢₀). By Lower Semi-Mosco convergence again, we have 𝑆(𝑢₀)⊆lim_𝑛inf𝑆(𝑢_𝑛). Also observe that assumption (ii) applied to the constant sequence ℎ_𝑛=𝑢₀, for all 𝑛∈𝑁, implies that int𝑆(𝑢₀)≠∅. From Lemma 2.5, it follows that if 𝑣∈int𝑆(𝑢₀), then there exist 𝑚∈𝑁 and 𝛿>0 such that int𝐵(𝑣,𝛿)⊆𝑆(𝑢_𝑛), for all 𝑛>𝑚. Thus, 𝑣∈𝑆(𝑢_𝑛) for 𝑛 sufficiently large. Notice the 𝐴 is monotone and the sequence (𝑢_𝑛)_𝑛 is an 𝛼-approximating sequence for (QVIP), then we have 𝐴𝑣,𝑢₀−𝑣=lim_𝑛𝐴𝑣,𝑢_𝑛−𝑣≤liminf_𝑛𝐴𝑢_𝑛,𝑢_𝑛−𝑣≤lim_𝑛𝜀_𝑛+𝛼2𝑢_𝑛−𝑣²=𝛼2𝑢₀−𝑣². (3.6) If 𝑣∈𝑆(𝑢₀)⧵int𝑆(𝑢₀), let (𝑣_𝑛)_𝑛 be a sequence converging to 𝑣, whose point belongs to a segment contained in int𝑆(𝑢₀). Since 𝑣_𝑛∈int𝑆(𝑢₀), for all 𝑛∈𝑁, one has 𝐴𝑣_𝑛,𝑢₀−𝑣_𝑛≤𝛼2𝑢₀−𝑣_𝑛². (3.7) Since the hemicontinuity of 𝐴, 𝐴𝑣,𝑢₀≤𝛼−𝑣2𝑢₀−𝑣²,∀𝑣∈𝑆𝑢₀. (3.8) It follows from Lemma 2.6 that 𝐴𝑢₀,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²≤0,∀𝑣∈𝑆𝑢₀, (3.9) then, by Lemma 2.7, we obtain that 𝑢₀ solves (QVIP). This completes the proof.

Now, we present a result in which assumption (ii) of above theorem is dropped, while the continuity assumption on the operator 𝐴 is strengthened.

Theorem 3.3. Let the following assumptions hold:(i)the set-valued mapping 𝑆 is nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converging to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converges to 𝑆(𝑢₀);(ii)the operator 𝐴 is (𝑠,𝑤)-continuous on 𝐾.
Then, (QVIP) is 𝛼-well-posed if and only if (3.4) holds.

Proof. The necessity follows from Theorem 3.1 and Lemma 2.7.
Conversely, let (𝑢_𝑛)_𝑛 be an 𝛼-approximating sequence for (QVIP) and (3.4) holds. From (3.4) and the proof of Theorem 3.2, we can obtain that (𝑢_𝑛)_𝑛 strongly converges to 𝑢₀, with 𝑢₀∈𝑆(𝑢₀). Since Lower Semi-Mosco convergence implies for every 𝑣∈𝑆(𝑢₀), there exists sequence (𝑣_𝑛)_𝑛 strongly converging to 𝑣 such that 𝑣_𝑛∈𝑆(𝑢_𝑛). Since the operator 𝐴 is (𝑠,𝑤)-continuous and (𝑢_𝑛)_𝑛 is an 𝛼-approximating sequence for (QVIP), we have 𝐴𝑢₀,𝑢₀−𝑣=lim_𝑛𝐴𝑢_𝑛,𝑢_𝑛−𝑣_𝑛≤lim_𝑛𝜀_𝑛+𝛼2𝑢_𝑛−𝑣_𝑛²=𝛼2𝑢₀−𝑣². (3.10) By Lemma 2.7, we obtain that 𝑢₀ solves (QVIP). This completes the proof.

Theorem 3.4. Let the following assumptions hold:
(i)the set-valued mapping 𝑆 is nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converges to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑢₀);(ii)for every converging sequence (ℎ_𝑛)_𝑛, there exists 𝑚∈𝑁, such thatint_𝑛≥𝑚𝑆ℎ_𝑛≠∅; (3.11)(iii)the operator 𝐴 is hemicontinuous and monotone on 𝐾.
Then, (QVIP) is L-𝛼-well-posed if and only if 𝐿_𝜀≠∅,∀𝜀>0,lim_𝜀→0diam𝐿_𝜀=0. (3.12)

Proof. Assume that (QVIP) is L-𝛼-well-posed, then it follows from the monotonicity of 𝐴 that ∅≠Γ≠𝐿_𝜀, for all 𝜀>0. It follows from Theorem 3.1(b) that the necessity can be completed.
Assume that (3.12) holds. Let (𝑢_𝑛)_𝑛 be an L-𝛼-approximating sequence for (QVIP), then there exists a sequence 𝜀_𝑛>0, with 𝜀_𝑛→0, such that 𝑢_𝑛∈𝐿_𝜀𝑛, for all 𝑛∈𝑁. Following the same argument as the proof of Theorem 3.1, it is easy to see lim_𝜀→0diam𝐿_𝜀=0 and Γ≠∅ imply that Γ is a singleton point set. In light of the assumption, (𝑢_𝑛)_𝑛 is a Cauchy sequence and strongly converges to a point 𝑢₀∈𝐾 and 𝑢₀∈𝑆(𝑢₀). Let 𝑣∈int𝑆(𝑢₀) and using Lemma 2.5, one has 𝑣∈𝑆(𝑢_𝑛), for 𝑛 sufficiently large. Then, we get 𝐴𝑣,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²=lim_𝑛𝐴𝑣,𝑢_𝑛−𝛼−𝑣2𝑢_𝑛−𝑣²≤lim_𝑛𝜀_𝑛=0. (3.13) If 𝑣∈𝑆(𝑢₀)⧵int𝑆(𝑢₀), let a sequence 𝑣_𝑛 converges to 𝑣, whose points belong to a segment contained in int𝑆(𝑢₀). Since 𝐴𝑣_𝑛,𝑢₀−𝑣_𝑛−𝛼2𝑢₀−𝑣_𝑛²≤0 (3.14) and the operator 𝐴 is hemicontinuous, one gets 𝐴𝑣,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²≤0. (3.15) According to Lemmas 2.6 and 2.7, 𝑢₀ is the solution of (QVIP).

Theorem 3.5. Let the following assumptions hold:(i)the set-valued mapping 𝑆 is nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converging to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converges to 𝑆(𝑢₀);(ii)the operator 𝐴 is (𝑠,𝑤)-continuous and monotone on 𝐾.
Then, (QVIP) is L-𝛼-well-posed if and only if (3.12) holds.

Proof. Assume (3.12) holds. Let (𝑢_𝑛)_𝑛 be an L-𝛼-approximating sequence for (QVIP), then there exists a sequence 𝜀_𝑛>0, with 𝜀_𝑛→0, such that (𝑢_𝑛)_𝑛⊂𝐿_𝜀𝑛, for all 𝑛∈𝑁. Since lim_𝜀→0diam𝐿_𝜀=0, (𝑢_𝑛)_𝑛 is a Cauchy sequence and converges to 𝑢₀. As the similar proof of Theorem 3.2, 𝑢₀∈𝑆(𝑢₀). Let 𝑣∈𝑆(𝑢₀). Since Lower Semi-Mosco convergence implies for every 𝑣∈𝑆(𝑢₀), there exists a sequence (𝑣_𝑛)_𝑛 converging to 𝑣, such that 𝑣_𝑛∈𝑆(𝑢_𝑛). Since 𝐴 is (𝑠,𝑤)-continuous and (𝑢_𝑛)_𝑛 is an L-𝛼-approximating sequence for (QVIP), one has 𝐴𝑣,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²=lim_𝑛𝐴𝑣_𝑛,𝑢_𝑛−𝑣_𝑛−𝛼2𝑢_𝑛−𝑣_𝑛²≤lim_𝑛𝜀_𝑛=0. (3.16) Applying Lemmas 2.6 and 2.7, we have that (QVIP) is L-𝛼-well-posed.
The necessity can be completed as Theorem 3.3.

4. 𝛼-Well-Posedness in the Generalized Sense

In this section, we introduce and investigate some metric characterizations of 𝛼-well-posedness in the generalized sense and 𝐿-𝛼-well-posedness in the generalized sense for (QVI).

Definition 4.1 (see [11]). Let (𝑋,𝑑) be a metric space and let 𝐴,𝐵 be nonempty subsets of 𝑋. The Hausdorff distance 𝐻(⋅,⋅) between 𝐴 and 𝐵 is defined by 𝐻(𝐴,𝐵)=max{𝑒(𝐴,𝐵),𝑒(𝐵,𝐴)}, (4.1) where 𝑒(𝐴,𝐵)=sup_𝑎∈𝐴𝑑(𝑎,𝐵) with 𝑑(𝑎,𝐵)=inf_𝑏∈𝐵‖𝑎−𝑏‖.

Definition 4.2 (see [11]). Let 𝐴 be a nonempty subset of 𝑋. The measure of non compactness 𝜇 of the set 𝐴 is defined by 𝜇(𝐴)=inf𝜀>0∶𝐴⊆^𝑛_𝑖=1𝐴_𝑖,diam𝐴_𝑖<𝜀,𝑖=1,2,…,𝑛, (4.2) where diam means the diameter of a set.

Theorem 4.3. Let the same assumptions be as in Lemma 2.7. Then, one has the following.(a)(QVIP) is 𝛼-well-posed in the generalized sense if and only if the solution set Γ of (QVIP) is nonempty compact and 𝑒(𝑄_𝜀,Γ)→0,as𝜀→0.(b)Moreover, if 𝐴 is pseudomonotone, then (QVIP) is 𝐿-𝛼-well-posed in the generalized sense if and only if the solution set Γ of (QVIP) is nonempty compact and 𝑒(𝐿_𝜀,Γ)→0,as𝜀→0.

Proof. We only prove (a), the proof of (b) is similar and is omitted here. Assume that (QVIP) is 𝛼-well-posed in the generalized sense, then the Γ is nonempty and compact. It follows from Lemma 2.7 that 𝑄_𝜀≠∅. Now we prove 𝑒(𝑄_𝜀,Γ)→0,as𝜀→0. Suppose by contradiction that there exists 𝛽>0,𝜀_𝑛→0, and 𝑤_𝑛∈𝑄_𝜀𝑛, such that 𝑑(𝑤_𝑛,Γ)≥𝛽. It follows from 𝑤_𝑛∈𝑄_𝜀𝑛 that (𝑤_𝑛)_𝑛 is an 𝛼-approximating sequence for (QVIP). (QVIP) is 𝛼-well-posedness in the generalized sense, then there exists a subsequence (𝑤_𝑛𝑘)_𝑘 of (𝑤_𝑛)_𝑛 strongly converging to a point of Γ. This contradicts 𝑑(𝑤_𝑛,Γ)≥𝛽. Thus, 𝑒(𝑄_𝜀,Γ)→0,as𝜀→0.
For the converse, let (𝑢_𝑛)_𝑛 be an 𝛼-approximating sequence for (QVIP), then 𝑢_𝑛∈𝑄_𝜀𝑛. It follows from 𝑒(𝑄_𝜀𝑛,Γ)→0 that there exists a sequence 𝑧_𝑛⊂Γ, such that 𝑑(𝑢_𝑛,𝑧_𝑛)→0. Since Γ is compact, there exists a subsequence (𝑧_𝑛𝑘)_𝑘 of (𝑧_𝑛)_𝑛 strongly converging to 𝑢₀∈Γ. Thus there exists the corresponding subsequence (𝑢_𝑛𝑘)_𝑘 of (𝑢_𝑛)_𝑛 strongly converging to 𝑢₀. Therefore, (QVIP) is 𝛼-well-posed in the generalized sense.

Theorem 4.4. (a) If (QVIP) is 𝛼-well-posed in the generalized sense, then 𝑄_𝜀≠∅,∀𝜀>0,lim_𝜀→0𝜇𝑄_𝜀=0. (4.3)
(b) If (4.3) and the following assumptions hold:
(i)the set-valued mapping 𝑆 is nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converges to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converging to 𝑆(𝑢₀);(ii)the operator 𝐴 is (𝑠,𝑤)-continuous on 𝐾,then, (QVIP) is 𝛼-well-posed in the generalized sense.

Proof. (a) Suppose that (QVIP) is 𝛼-well-posed in the generalized sense. So 𝑄_𝜀≠∅, for all 𝜀>0. By Theorem 4.3(a), Γ is nonempty compact and 𝑒(𝑄_𝜀,Γ)→0,as𝜀→0. For any 𝜀>0, we have 𝐻𝑄_𝜀,Γ=max𝑒𝑄_𝜀,Γ,𝑒Γ,𝑄_𝜀=𝑒𝑄_𝜀,Γ, (4.4)and since Γ is compact, 𝜇(Γ)=0. For every 𝑛∈𝑁, the following relation holds [14]: 𝜇𝑄_𝜀≤2𝐻𝑄_𝜀,Γ+𝜇(Γ)=2𝐻𝑄_𝜀,Γ=2𝑒𝑄_𝜀,Γ. (4.5) It follows from 𝑒(𝑄_𝜀,Γ)→0,as𝜀→0, that lim_𝜀→0𝜇(𝑄_𝜀)=0.
(b) Assume that (4.3) holds. Then, for any 𝜀>0, cl(𝑄_𝜀) is nonempty closed and increasing with 𝜀>0. By (4.3), lim_𝜀→0𝜇(cl(𝑄_𝜀))=lim_𝜀→0𝜇(𝑄_𝜀)=0, where cl(𝑄_𝜀) is the closure of 𝑄_𝜀. By the generalized Cantor theorem [11, page 412], we know that lim_𝜀→0𝐻cl𝑄_𝜀,Δ=0,as𝜀⟶0, (4.6) where Δ=⋂_𝜀>0cl(𝑄_𝜀) is nonempty compact.
Now we show that Γ=Δ. (4.7) It follows from Lemma 2.7 that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let 𝑢₀∈Δ. Then, 𝑑(𝑢₀,𝑄_𝜀)=0 for every 𝜀>0. Given 𝜀_𝑛>0, 𝜀_𝑛→0, for every 𝑛, there exists 𝑢_𝑛∈𝑄_𝜀𝑛 such that 𝑑(𝑢₀,𝑢_𝑛)<𝜀_𝑛. Hence, 𝑢_𝑛→𝑢₀ and 𝑑𝑢_𝑛,𝑆𝑢_𝑛≤𝜀_𝑛,𝐴𝑢_𝑛,𝑢_𝑛−𝑣≤𝜀_𝑛+𝛼2𝑢_𝑛−𝑣²,∀𝑣∈𝑆𝑢_𝑛. (4.8)(4.9) It follows from (4.8), 𝑢_𝑛→𝑢₀, and the proof of Theorem 3.2 that 𝑢₀∈𝑆(𝑢₀).
Since Lower Semi-Mosco convergence implies that, for every 𝑣∈𝑆(𝑢₀), there exists a sequence 𝑣_𝑛∈𝑆(𝑢_𝑛), for all 𝑛∈𝑁, such that lim_𝑛𝑣_𝑛=𝑣 in the strongly topology.
Since the operator 𝐴 is (𝑠,𝑤)-continuous on 𝐾, hence 𝐴𝑢₀,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²=lim_𝑛𝐴𝑢_𝑛,𝑢_𝑛−𝑣_𝑛−𝛼2𝑢_𝑛−𝑣_𝑛²≤lim_𝑛𝜀_𝑛=0. (4.10) By Lemma 2.7, we know 𝑢₀∈Γ. Thus, Δ⊆Γ. It follows from (4.6) and (4.7) that lim_𝜀→0𝑒(𝑄_𝜀,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(a) that (QVIP) is 𝛼-well-posed in the generalized sense. The proof is completed.

Theorem 4.5. Let 𝐾 be a nonempty, compact, and convex subset of 𝐸, let the set-valued mapping 𝑆 be nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converging to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converges to 𝑆(𝑢₀), and the operator 𝐴 is (𝑠,𝑤)-continuous on 𝐾. Then, (QVIP) is 𝛼-well-posed in the generalized sense.

Proof. Let (𝑢_𝑛)_𝑛 be an 𝛼-approximating sequence for (QVIP). Since the set 𝐾 is compact, there exists subsequence (𝑢_𝑛𝑘)_𝑘 of (𝑢_𝑛)_𝑛 strongly converging to a point 𝑢₀∈𝐾. Reasoning as in Theorem 3.3, we get 𝑢₀∈𝑆(𝑢₀) and 𝑢₀ solves (QVIP). Therefore, (QVIP) is 𝛼-well-posed in the generalized sense.

Theorem 4.6. Let the following assumptions hold:
(i)the set-valued mapping 𝑆 is nonempty convex-valued, and, for each sequence (𝑢_𝑛)_𝑛 in 𝐾 converging to 𝑢₀, the sequence (𝑆(𝑢_𝑛))_𝑛 Lower Semi-Mosco converges to 𝑆(𝑢₀);(ii)the operator 𝐴 is (𝑠,𝑤)-continuous and monotone on 𝐾.
Then, (QVIP) is L-𝛼-well-posed in the generalized sense if and only if 𝐿_𝜀≠∅,∀𝜀>0,lim_𝜀→0𝜇𝐿_𝜀=0. (4.11)

Proof. Assume that (QVIP) is L-𝛼-well-posed in the generalized sense. It follows from Lemma 2.7 and the monotonicity of 𝐴 that Γ⊂𝐿_𝜀, for all 𝜀>0. And so 𝐿_𝜀≠∅, for each 𝜀>0. By Theorem 4.3(b), we can get 𝑒(𝐿_𝜀,Γ)→0 as 𝜀→0. From the proof of Theorem 4.4, we also obtain 𝜇𝐿_𝜀≤2𝐻𝐿_𝜀,Γ+𝜇(Γ)=2𝐻𝐿_𝜀,Γ=2𝑒𝐿_𝜀,Γ. (4.12) Thus, lim_𝜀→0𝜇(𝐿_𝜀)=0.
Conversely, assume (4.11) holds. Then, for any 𝜀>0, cl(𝐿_𝜀) is nonempty closed and increasing with 𝜀>0. By (4.11), lim_𝜀→0𝜇(cl(𝐿_𝜀))=lim_𝜀→0𝜇(𝐿_𝜀)=0, where cl(𝐿_𝜀) is the closure of 𝐿_𝜀. By the generalized Cantor theorem [11, page 412], we know that lim_𝜀→0𝐻cl𝐿_𝜀,Δ=0,as𝜀⟶0, (4.13) where Δ=⋂_𝜀>0cl(𝐿_𝜀) is nonempty compact.
Now we show that Γ=Δ. (4.14) It follow from Lemma 2.7 and the monotonicity of 𝐴 that Γ⊆Δ. So we need to prove that Δ⊆Γ. Indeed, let 𝑢₀∈Δ. Then 𝑑(𝑢₀,𝐿_𝜀)=0 for every 𝜀>0. Given 𝜀_𝑛>0, 𝜀_𝑛→0, for every 𝑛, there exists 𝑢_𝑛∈𝐿_𝜀𝑛 such that 𝑑(𝑢₀,𝑢_𝑛)<𝜀_𝑛. Hence, 𝑢_𝑛→𝑢₀ and 𝑑𝑢_𝑛,𝑆𝑢_𝑛≤𝜀_𝑛,𝐴𝑣,𝑢_𝑛−𝑣≤𝜀_𝑛+𝛼2𝑢_𝑛−𝑣²,∀𝑣∈𝑆𝑢_𝑛. (4.15)(4.16)
It follows from (4.15), 𝑥_𝑛→𝑥₀, and the proof of Theorem 3.2 that 𝑢₀∈𝑆(𝑢₀).
Since 𝑆(𝑢_𝑛) Lower Semi-Mosco converges to 𝑆(𝑢₀), for every 𝑣∈𝑆(𝑢₀), there exists a sequence 𝑣_𝑛∈𝑆(𝑢_𝑛), for all 𝑛∈𝑁, such that lim_𝑛𝑣_𝑛=𝑣 in the strong topology.
Since the operator 𝐴 is (𝑠,𝑤)-continuous on 𝐾, hence 𝐴𝑣,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²=lim_𝑛𝐴𝑣_𝑛,𝑢_𝑛−𝑣_𝑛−𝛼2𝑢_𝑛−𝑣_𝑛²≤lim_𝑛𝜀_𝑛=0. (4.17) By Lemma 2.6 we know that 𝑢₀∈𝑆(𝑢₀), such that 𝐴𝑢₀,𝑢₀−𝛼−𝑣2𝑢₀−𝑣²≤0,∀𝑣∈𝑆𝑢₀. (4.18) It follow from Lemma 2.7 that 𝑢₀∈Γ. Thus, Δ⊆Γ. It follows from (4.13) and (4.14) that lim_𝜀→0𝑒(𝐿_𝜀,Γ)=0. It follows from the compactness of Γ and Theorem 4.3(b) that (QVIP) is L-𝛼-well-posed in the generalized sense. The problem is completed.

Remark 4.7. It is easy to see that if 𝛼=0, then by the main results in our paper, we can recover the corresponding results in [9] with the weaker condition that 𝑆(𝑥_𝑛) Lower Semi-Mosco converges to 𝑆(𝑥₀) instead of the condition that 𝑆 is (𝑠,𝑤)-closed and (𝑠,𝑤)-subcontinuous, and (𝑠,𝑠)-lower semicontinuous.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11171363 and Grant no. 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01) and the Project of the Third Batch Support Program for Excellent Talents of Chongqing City High Colleges.